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\begin{document}

\title{Analitik Geometri (MAT 104)\\
Ara S\i nav\i\ \color{blue}\c C\"oz\"umleri}
\date{6 Nisan 2014}
\author{David Pierce}
%\abstract{}
\maketitle
\thispagestyle{empty}

\begin{problem}
$ab=de$ ve $ac=df$ ise
  \begin{equation*}
b:c::e:f
  \end{equation*}
  orant\i s\i n\i\ kan\i tlay\i n.
(Bundan \"once kan\i tlad\i\u g\i m\i z teoremleri kullanabilirsiniz.)
\end{problem}

\begin{solution}
Varsay\i mdan
\begin{align*}
a:d&::e:b,&a:d&::f:c,
\end{align*}
dolay\i s\i yla
\begin{gather*}
e:b::f:c,\\
e:f::b:c,
\end{gather*}
yani $b:c::e:f$.
\end{solution}

\newpage
\begin{problem}
  $0<\ell<2a$ ise
  \begin{align*}
    2b^2&=\ell a,&c&=a-\sqrt{a^2-b^2},&d&=\frac{ac}{\sqrt{a^2-b^2}}
  \end{align*}
olsun.
Sadece $\ell$ ve $a$ uzunluklar\i n\i\ kullanarak 
$\displaystyle\frac{c^2}{d^2}-1$ fark\i n\i\ 
en basit bi\c cimde yaz\i n.
\end{problem}

\begin{solution}
  \begin{align*}
    \frac{c^2}{d^2}-1
&=\frac{c^2}{\left(\displaystyle\frac{ac}{\sqrt{a^2-b^2}}\right)^2}-1\\
&=\frac{a^2-b^2}{a^2}-1
=-\frac{b^2}{a^2}
=-\frac{\ell}{2a}.
  \end{align*}
\end{solution}

\newpage

\begin{problem}
  Dik $xy$ eksenlerine g\"ore,
birim uzunlu\u gunun se\c cildi\u gi durumda,
tabloyu doldurun
ve koni kesitlerini \c cizin.
\begin{center}\renewcommand{\arraystretch}{1.5}
  \begin{tabular}{r|c|c}
    &$16x^2+256=9y^2+160x$&$8x+y^2+8y=0$\\\hline
ad&\color{blue}hiperbol&\color{blue}parabol\\\hline
k\"o\c se(ler)&\color{blue}$(8,0)$, $(2,0)$&\color{blue}$(2,-4)$\\\hline
odak(lar)&\color{blue}$(10,0)$, $(0,0)$&\color{blue}$(0,-4)$\\\hline
eksen&\color{blue}$y=0$&\color{blue}$y+4=0$\\\hline
  \end{tabular}
\end{center}
\end{problem}

\begin{solution}
  $\begin{gathered}[t]
  \begin{aligned}[t]
&\phantom{{}\iff{}}16x^2+256=9y^2+160x\\
&\iff 16(x^2-10x+25)-400+256=9y^2\\
&\iff 16(x-5)^2-9y^2=144\\
&\iff\frac{(x-5)^2}9-\frac{y^2}{16}=1,
  \end{aligned}\\
  \begin{aligned}
8x+y^2+8y=0
&\iff y^2+8y+16=16-8x\\
&\iff(y+4)^2=-8(x-2).
  \end{aligned}
  \end{gathered}$

\psset{unit=3mm,linecolor=blue}
  \begin{pspicture}(-7,-8)(9,9)
    \psplot37{4 x x mul 9 div 1 sub sqrt mul}
    \psplot37{4 x x mul 9 div 1 sub sqrt mul neg}
    \psplot{-7}{-3}{4 x x mul 9 div 1 sub sqrt mul}
    \psplot{-7}{-3}{4 x x mul 9 div 1 sub sqrt mul neg}
\psline{->}(-7,0)(9,0)
\psline{->}(-5,-8)(-5,8)
\psdots(-5,0)(-3,0)(3,0)(5,0)
\uput[dr](5,0){$10$}
\uput[dl](3,0){$8$}
\uput[dr](-3,0){$2$}
  \end{pspicture}
\hfill
\begin{pspicture}(-8,-8)(8,8)
  \psplot{-8}0{x 8 mul neg sqrt}
  \psplot{-8}0{x 8 mul neg sqrt neg}
\psline{->}(-8,4)(8,4)
\psline{->}(-2,-8)(-2,8)
\psline(-8,0)(0,0)
\psdots(0,0)(-2,0)
\uput[r](0,0){$(2,-4)$}
\end{pspicture}%
\begin{comment}
\begin{center}%\renewcommand{\arraystretch}{1.5}
\footnotesize
  \begin{tabular}{r|c|c}
    &$16x^2+256=9y^2+160x$&$8x+y^2+8y=0$\\\hline
ad&hiperbol&parabol\\\hline
k\"o\c se(ler)&$(-8,0)$, $(-2,0)$&$(2,-4)$\\\hline
odak(lar)&$(10,0)$, $(0,0)$&$(0,-4)$\\\hline
eksen&$y=0$&$y+4=0$\\\hline
  \end{tabular}
\end{center}
\end{comment}
\end{solution}

\newpage
\begin{problem}
\c Sekillerde
\begin{compactitem}
  \item
$BAC$ (veya $ABC$) e\u grisi, \c cap\i\ $AD$ ve k\"o\c sesi $A$ olan parabol,
\item
$BD$ ve $CE$ ordinat,
\item
$FA=AD$, ve 
\item
$BG\parallel FE$,
$CH\parallel BF$ 
\end{compactitem}
olsun.
A\c sa\u g\i daki i\c saretli uzunluklar tan\i mlans\i n:
\begin{align*}
  &
  \begin{gathered}[t]
    \ydp{AD}=a,\\
\ydp{DB}=b,\\
\ydp{FB}=c,
  \end{gathered}&
&\begin{gathered}[t]
 \ydp{AE}=x,\\
\ydp{EC}=y,   
  \end{gathered}&
&\begin{gathered}[t]
 \ydp{BH}=s,\\
\ydp{HC}=t,   
  \end{gathered}&
&\phantom{ABC=dddddddddd}
\end{align*}

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\psline(-0.5625,0)(-0.1875,0.75)%(0.1825,1.5) % new y axis
\parametricplot{-0.375}{0.75}{t t mul t sub
                       t} % parabola x = y^2 - y 
\psline(-0.1825,0.75)(0.5625,0) % old ordinate of new origin
%\psline(0.0625,0.75)(0.3125,1.25) % new ordinate of point
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\uput[dl](0,0){$A$}
\uput[u](-0.1875,0.75){$B$}
\uput[d](0.515625,-0.375){$C$}
\uput[d](0.5625,0){$D$}
\uput[d](-0.5625,0){$F$}
\color{blue}\psset{linecolor=blue,linestyle=dashed,linewidth=1.6pt}
\pspolygon(0.515625,-0.375)(-0.609375,0.75)(1.078125,0.75)
\uput[u](-0.609375,0.75){$G$}
\uput[u](1.078125,0.75){$H$}
\uput[ur](0.140625,0){$E$}
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\hfill
\begin{pspicture}(-0.5625,-0.4)(1.25,0.8)
\psline(-0.5625,0)(1.5625,0) % old x axis
%\psline(0,0)(-1.25,1.25) % old y axis
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\psline(-0.5625,0)(-0.1875,0.75)%(0.1825,1.5) % new y axis
\parametricplot{0}{1.25}{t t mul t sub
                       t} % parabola x = y^2 - y 
\psline(-0.1825,0.75)(0.5625,0) % old ordinate of new origin
\psline(0.0625,0.75)(0.3125,1.25) % new ordinate of point
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\uput[ul](-0.1875,0.75){$B$}
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\uput[d](1.5625,0){$E$}
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\uput[dr](0.0625,0.75){$H$}
\uput[ur](0.8125,0.75){$G$}
  \end{pspicture}

\begin{enumerate}[a)]
\item 
Soldaki \c sekli tamamlay\i n.
\item
K\"u\c c\"uk harfleri kullanarak 
$\ydp{GH}=\underline{\phantom{MM}\text{\color{blue}\large$-x+a+s$}\phantom{MM}}$?
\item
Sadece $a$, $b$, $c$, $s$, ve $t$ uzunluklar\i n\i\ kullanarak 
$x$ uzunlu\u gunu yaz\i n.
\end{enumerate}
\end{problem}

\begin{solution}
$    \ydp{GH}
=\ydp{GB}+\ydp{BH}
=\ydp{ED}+\ydp{BH}
=\ydp{EA}+\ydp{AD}+\ydp{BH}=-x+a+s$,

$\ydp{GH}:\ydp{HC}::\ydp{DF}:\ydp{FB}$,
$\displaystyle\frac{-x+a+s}t=\frac{-2a}c$,
$x=\displaystyle\frac{2at}c+a+s$.
\end{solution}
\end{document}
